$f$ is not linear in this case. It is affine. Linear functions from $\mathbb{R}$ to $\mathbb{R}$ look like $f(x)=cx$ where $c$ is a fixed constant. This is assuming that $f$ is continuous of course.
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Daniel MontealegreApr 2 '12 at 23:05

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Linear functions in the sense of $f(x)=ax+b$ are functions whose graph is a straight line. This notion coincides with linear in the sense of additivity if and only if $b=0$.
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Asaf KaragilaApr 2 '12 at 23:07

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You are right, the terminologies are almost contradictory. Your function $f(x)=7x+3$ is fairly often called a linear function. But, as your calculation, it does not yield a linear transformation.
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André NicolasApr 2 '12 at 23:12

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A function $f(x)$ is linear in one sense if it is of the form $f(x)=ax+b$ for constants $a$ and $b$. This simply means that it is a polynomial of degree less than $2$. In graphical terms, it means that the graph is a straight line, hence the name linear.

A function $f(x)$ is linear in the other sense if it satisfies the condition $$f(ax+by)=af(x)+bf(y)\;.$$

The two meanings are unrelated. In particular, a linear function in the first sense is linear in the second sense if and only if $b=0$. In your example $b=3$, so while your function is linear in the first sense, it is not linear in the second sense.

@Bill: I don’t consider the statement false at the level at which the question was posed. And I consider it very bad pædagogy to overwhelm students with information that is more likely to be confusing than helpful.
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Brian M. ScottApr 3 '12 at 0:07